Pricing barrier and lookback options using finite difference numerical methods
Umeorah, Nneka Ozioma
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This research work focuses on the estimation of barrier and lookback option prices using finite difference numerical methods. Here, we aim at approximating the fair prices of the zero rebate up-and-out and down-and-out knock out barrier options, as well as the fixed strike lookback options. Simulation and finite difference techniques will be used to approximate these prices. The Monte-Carlo simulation, the antithetic Monte- Carlo simulations and the Crank-Nicolson approach will be specifically employed on the barrier options. Other finite difference methods like the implicit and the explicit method will be discussed but the Crank-Nicolson method will be employed in the numerical valuations owing to its accuracy in comparison to others. Next, the fixed strike lookback option prices will be estimated using the Monte-Carlo and the antithetic Monte-Carlo simulation methods. An extended version of the Black-Scholes model will be used in the valuation of their exact prices owing to their exotic nature. The Monte-Carlo and the antithetic Monte-Carlo methods are next employed to simulate the values of these option prices. The resulting prices will be compared to the exact fair prices and this will be followed by some error analysis. From the findings, the antithetic method gave the best option price estimate in comparison to the ordinary Monte-Carlo method when the simulation approach was used. It will also be observed that the Monte-Carlo simulation had a slow rate of convergence as a result of higher variances of the estimate from the true solution. Hence, such ineficiency was curbed by the introduction of antithetic Monte-Carlo simulation which had smaller variances of the estimate, and this in turn gave a better estimate. Furthermore, it will also be observed that the Crank-Nicolson method converged faster with increase in the discretisation steps of the underlying asset and the time.